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TchrWill
Advanced Member

USA
79 Posts

Posted - 07/08/2013 :  15:44:09  Show Profile  Reply with Quote
The digits of a 5 digit number ABCDE have the following relationships:

1--The first two digits are the square of the third digit.
2--The fourth and fifth digits are the sum of the second and third digits.
3--The third digit is twice the second digit.
4--The fifth digit is one half the second digit and one third the first digit.
5--The sum of the five digits is 21.
6--What is the 5 digit number?
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someguy
Advanced Member

Canada
143 Posts

Posted - 07/28/2013 :  13:43:23  Show Profile  Reply with Quote
Properties 1 and 3 tell you how to start forming the number in question.

Properties 1 and 3 say AB is C^2 and C=2*B. We also know C is an integer between 0 and 9.
00 (0 is 2*0)
01 (1 is not 2*1)
04 (2 is not 2*4)
09 (3 is not 2*9)
16 (4 is not 2*6)
25 (5 is not 2*5)
36 (6 is not 2*6)
49 (7 is not 2*9)
64 (8 is 2*4)
81 (9 is not 2*1)

ABC is either 000 or 648

Having the 5 digits sum to 21 rules out 000.

The number is 648DE.

The second and third digits sum to 12.

The number is 64812

Note that property 4 was not used, but is satisfied.

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TchrWill
Advanced Member

USA
79 Posts

Posted - 07/29/2013 :  13:59:33  Show Profile  Reply with Quote
quote:
Originally posted by someguy

Properties 1 and 3 tell you how to start forming the number in question.

Properties 1 and 3 say AB is C^2 and C=2*B. We also know C is an integer between 0 and 9.
00 (0 is 2*0)
01 (1 is not 2*1)
04 (2 is not 2*4)
09 (3 is not 2*9)
16 (4 is not 2*6)
25 (5 is not 2*5)
36 (6 is not 2*6)
49 (7 is not 2*9)
64 (8 is 2*4)
81 (9 is not 2*1)

ABC is either 000 or 648

Having the 5 digits sum to 21 rules out 000.

The number is 648DE.

The second and third digits sum to 12.

The number is 64812

Note that property 4 was not used, but is satisfied.





Nice approach someguy. I have concluded that I tend to take a longer algebraic path. But, that is what makes recreational math interesting.

From the given inforamtion:
1--10A + B = C^2
2--10D + E = B + C
3--C = 2B
4--E = B/2 = A/3
5--Since 10A + B is a square, the only 2 digit numbers possible are 16, 25, 36, 49, 64, and 81.
6--This makes the ABC possibilities 164, 255, 366, 497, 648 and 819.
7--Only 648 satisfies the C = 2B requirement and therefore ABC = 648.
8--WIth E = B/2 or A/3, E muat be 2 for 648D2.
9--Since 10D + E = B + C, 10D + 2 = 4 + 8 making D = 1.
10--Thus, the number we seek is 64,812 without using given fact #5.
11--Checking: 6+4+8+1+2 = 21.

Alternatively
1--10A + B = C^2
2--10D + E = B + C
3--C = 2B
4--E = B/2 = A/3
5--A + B + C + D + E = 21
6--3E + 2E + 4E + (B + C - E)/10 + E = 21
7--3E + 2E + 4E + 2E/10 + 4E/10 - E/10 + E = 21
8--Times 10, 30E + 20E + 40E + 2E + 4E - E + 10E = 210
9--Collecting, 105E = 210 making E = 2.
10--B = 2E making B = 4
11--A = 3E making A = 6.
12--With AB = 64, C = 8.
13--Therefore, D = 21 - 6 - 4 - 8 - 2 = 1 for 64,812
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